Ok, I agree they do not really mean the same. The first one reminded me of the second - and i do think there might be a link (in the sense that you should not wanting too much, because you might lose too much). But I was mainly wondering about the translation of the bird proverb !

As for the German "Der Spatz (Ein Sperling) in der Hand is besser als eine Taube auf dem Dache”, the translation is, I think: a/ the sparrow in the hand is better than a pigeon (dove) on the roof(better a sparrow in the hand than...)

The earliest English version of the proverb is from the Bible and was translated into English in Wycliffe's version in 1382, although Latin texts have it from the 13th century:

Ecclesiastes IX - A living dog is better than a dead lion.
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Alternatives that explicitly mention birds in hand come later. The earliest of those is in Hugh Rhodes' The boke of nurture or schoole of good maners, circa 1530:

"A byrd in hand - is worth ten flye at large."
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John Heywood, the 16th century collector of proverbs, recorded another version in his ambitiously titled A dialogue conteinyng the nomber in effect of all the prouerbes in the Englishe tongue, 1546:

"Better one byrde in hande than ten in the wood."
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The Bird in Hand was adopted as a pub name in the UK in the Middle Ages and there are still many pubs of that name there. This refers back to mediaeval falconry where a bird in the hand (the falcon) was certainly worth more than two in the bush (the prey).